A Game Between Two Identical Dubins Cars: Evading a Conic Sensor in Minimum Time

TL;DR

This paper studies a surveillance pursuit-evasion game between two identical Dubins cars where the pursuer carries a semi-infinite conic field of view with half-angle $\phi_d$ and aims to maximize the time the evader remains inside the cone. Using Isaacs' differential-game framework and retro-time integration, it derives time-optimal endgame strategies and reveals two key singular surfaces: a Transition Surface and an Evader's Universal Surface, with analytical trajectory expressions and numerical methods needed to identify switches. A notable finding is that the barrier constructed from the usable part boundary can partially lie outside the playable space, suggesting that additional singular surfaces are required to fully characterize the game. Simulations illustrate the endgame dynamics, including transitions between primary solutions and EUS tributaries, highlighting the practical complexity of surveillance with conic sensors on nonholonomic agents. Overall, the work lays foundational analytical insights for cone-FOV pursuit-evasion with Dubins dynamics and points to directions for discovering missing barrier structures.

Abstract

A fundamental task in mobile robotics is keeping an intelligent agent under surveillance with an autonomous robot as it travels in the environment. This work studies a theoretical version of that problem involving one of the most popular vehicle platforms in robotics. In particular, we consider two identical Dubins cars moving on a plane without obstacles. One of them plays as the pursuer, and it is equipped with a limited field-of-view detection region modeled as a semi-infinite cone with its apex at the pursuer's position. The pursuer aims to maintain the other Dubins car, which plays as the evader, as much time as possible inside its detection region. On the contrary, the evader wants to escape as soon as possible. In this work, employing differential game theory, we find the time-optimal motion strategies near the game's end. The analysis of those trajectories reveals the existence of at least two singular surfaces: a Transition Surface (also known as a Switch Surface) and an Evader's Universal Surface. We also found that the barrier's standard construction produces a surface that partially lies outside the playing space.

Figures (16)

• Figure 1: The pursuer and the evader are represented by the blue and red dots, respectively.
• Figure 2: Representation of the Usable Part (UP), blue region, and its boundary (BUP), magenta curves, in the reduced space for $\phi_d=40^\circ$. The Usable Part Line (UPL) corresponds to the yellow line. The gray rectangles represent the semi-infinite cone as $\theta$ varies from $0$ to $2\pi$. The image shows views from an observer located in front of the semi-infinite cone.
• Figure 3: Primary solution (red curves) for $\phi_d=40^\circ$. The trajectories start from the UP and continue backward in time until they hit the boundary of the detection region (gray region) again, or the pursuer switches its control. Note that for all configurations in the gray region $\phi=\phi_d$, however, the evader cannot immediately escape, i.e., they do not belong to the UP.
• Figure 4: An example of the Evader's Universal Surface (black curves) and its tributary trajectories for $\phi_d=40^\circ$. The green and yellow colors indicate the trajectories at each side of the EUS. The evader applies a particular control at each side, i.e., $\nu_e^*=-1$ or $\nu_e^*=1$. In retro-time, the tributary trajectories continue until they hit the boundary of the detection region or the pursuer switches its control.
• Figure 5: An example of the trajectories reaching the Transition Surface (orange curves) for $\phi_d=40^\circ$. The Transition surface corresponds to the points where the orange curves meet the primary solution (red curves).